The Distance Formula & Equations of Circles

The Distance Formula&Equations of Circles Proving the distance formula Examples of finding the distance between two points Proving equation of circles Examples of equations of circles

Proving the Distance Formula • If you have two points and are trying to find the distance between those points, you use the Pythagorean Theorem. • So, using the Pythagorean Theorem (a2 + b2 = c2), you can assume that α2 + β2 = µ2 • So, the distance from P to Q is written: • d(P,Q) = P (x2,y2) µ β Q α (x1,y1) (x2,y1) Whereα = x2-x1 andβ = y2-y1 and µ = distance from point P to point Q

Examples • Find the distance between the points • (2,4) & (3,-5) sqrt[(3-2)2 + (-5-4)2] = sqrt(1 + 81) = sqrt(82) (2,4) (3,-5)

Examples cont. • Find the distance between the points • (-5,-5) & (4,-3) sqrt[(-5-4)2 + (-5-(-3))2] = sqrt(81 + 4) = sqrt(85) (4,-3) (-5,-5)

Examples cont. • Find the distance between the points • (1,4) & (7,4) sqrt[(1-7)2 + (4-4)-2] = sqrt(36 + 0) = sqrt(36) = 6 (7,4) (1,4)

Equation of Circles • With a circle with a radius r, centered at point P, all points that are r units away from P make the circle. • If the coordinates of P are (h,k), you can use the distance formula to get • If we square this, we get the standard equation of a circle: (x-h)2 + (y-k)2 = r r r P (h,k)

Examples • Draw the graph of the equation (x-2)2 + (y+2)2 = 9 (h,k) = (2,-2) r = 3

Examples cont. • Draw the graph of the equation (x+3)2 + (y+4)2 = 4 (h,k) = (-3,-4) r = 2

The Distance Formula & Equations of Circles